Thursday, April 11, 2019

A Travelers Definition of Infinity

Now I am not a mathematician nor is the upcoming subject the usual fare for this blog, but here goes anyway.

The concept of infinity has bothered me for a long time. It first really bothered when I was learning calculus and heard that there is always a very small remainder in the typical calculus answer. I foolishly allowed this trivial observation to undermine my confidence in the mathematical strength of calculus. In my mind, I was learning a failed technique that was not perfect. Now, some sixty years later, I am confident that any calculus calculation can be as accurate as we desire, despite being incapable of achieving absolute perfection.

So what might an infinitely small error look like?

I was recently on a long drive in the Moses Lake, Washington area, traveling at the speed limit, when a container truck passed me. After complaining silently about his excessive speed, I began to wonder about the included angle made by the two sides of the rapidly disappearing truck. The angle was large when the truck was close but rapidly became smaller as it increased distance. The angle would become infinitely small at some distance (I thought) but no, it would never become zero. So, I wondered, what is infinity?

This must be a case of defined infinity. Here, "infinitely small" must be a shorthand term for 'too small to worry about'. We could always find an included angle with a value, even if the truck was located on a planet orbiting a distant star. The result would be real but how could it possibly have a physical influence on our local reality?

With this example in my background, I am free to think about the equation

Y = A_infinity x 1/B_infinity.

Does Y equal 1, zero, or infinity? It would depend upon the equality of the two infinities.

So there you have it. A travelers definition of infinity.

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